Collaborative Number Theory Seminar at the CUNY Graduate Center

Co-organizers: Gautam Chinta, Brooke Feigon, Maria Sabitova, Lucien Szpiro.

The seminar currently meets Fridays 4:00 to 5:30 PM in Room 4422. The CUNY Graduate Center is located on Fifth Avenue, on the east side of the street, between 34th and 35th Streets in midtown Manhattan. For further information, please contact Maria Sabitova.

Fall 2012 Schedule:

September 7: Robert Osburn (University College Dublin, Ireland)

Title: The mock and mixed mock modularity of q-hypergeometric series

Abstract: An intriguing and almost completely unsolved problem is to understand the overlap between classes of q-hypergeometric series and modular forms. This challenge was the subject of George Andrews' plenary address at the Millennial Conference on Number Theory and has its origin in Ramanujan's last letter to G.H. Hardy on January 12, 1920 whereby 17 mock theta functions were introduced. We discuss recent work concerning the explicit construction of new individual examples and infinite families of mock theta functions (in the modern sense of Zagier). Additionally, we discuss various ways to produce q-hypergeometric series which are mixed mock modular forms. This is joint work with Jeremy Lovejoy (Paris 7).

September 14: No meeting this week.

September 21: No meeting this week.

September 28: No meeting this week.

October 5: No meeting this week.

October 12: Dubi Kelmer (Boston College)

Title: The distribution of modular knots with a prescribed linking number.

Abstract: A closed geodesic on (the unit tangent bundle of) the modular surface gives rise to a knot on the 3-sphere with a trefoil knot removed, and one can compute the linking number of such a knot with the trefoil knot. Using the Selberg trace formula for modular forms with real weights, it is possible to compute the asymptotics for the number of closed geodesics with bounded length and a prescribed linking number.  In this talk, I will show how, by using combination of  this method and the thermodynamic formalism for the Gauss map, it is possible to show that this thin set of closed geodesics become equidistributed with respect to the volume measure.

October 19: David Goldberg (Purdue University)

Title: Exceptional representations and Whittaker models for higher metaplectic groups

Abstract: Kazhdan and Patterson developed a theory of metaplectic forms for (almost) arbitrary metaplectic covers of GL(r). A crucial part of this is the local theory, in which they defined and studied exceptional representations, including determining the space of their Whittaker models. In joint work with S. Friedberg and D. Szpruch we are developing a similar theory for n-fold covers of classical groups and similitude groups. In the case of classical groups, we show, similar to the case of GL(n) that the existence of Whittaker models is connected with counting of orbits of a group action on a lattice, and involves Gauss sums. We discuss this and its possible applications to number theory.

October 26: No meeting this week.

November 2: No meeting this week.

November 9: YoungJu Choie (POSTECH -- Pohang University of Science and                            Technology, Korea)

Title: Sturm type theorem for Siegel modular forms of genus 2 modulo p

Abstract: Suppose that f is an elliptic modular form with integral coefficients. Sturm obtained in 1987 bounds for a non-negative integer n such that every Fourier coefficient of f vanishes modulo a prime p if the first n Fourier coefficients of f are zero modulo p. In the present note we study analogues of Sturm's bounds for Siegel modular forms of genus 2. As an application we study congruences involving an analogue of Atkin's U(p)-operator for the Fourier coefficients of Siegel modular forms of genus 2. This is a joint work with D. Choi and T. Kikuta.

November 16: David Savitt (University of Arizona)

Title: The Buzzard-Diamond-Jarvis conjecture for unitary groups

Abstract:  We will discuss the proof of the weight part of Serre's conjecture for rank two unitary groups in the unramified case (that is, the Buzzard-Diamond-Jarvis conjecture for unitary groups).  This is joint work with Toby Gee and Tong Liu.  More precisely, we prove that any Serre weight which occurs is a predicted weight; this completes the analysis begun by Barnet-Lamb, Gee, and Geraghty, who proved that all predicted weights occur.  Our methods are purely local, using Liu's theory of (phi,G-hat) modules to determine the possible reductions mod p of certain two-dimensional crystalline Galois representations.

November 23: Thanksgiving

November 30: No meeting this week.

December 7: Krzysztof Klosin (CUNY Queens College)

Title: Congruences among hermitian modular forms and bounds on Selmer groups

Abstract: We will explain a construction of congruences between the so-called Maass lift (which associates a modular form on the unitary group U(2,2) to an elliptic modular form) and modular forms on U(2,2) which do not arise as Maass lifts. Such a congruence leads to evidence for the Bloch-Kato conjecture for the symmetric square motive of a modular form. If time permits we will also discuss a recent joint work with Jim Brown on extending this construction to the case of p-adic families.